2) Gradient distortion methods: This dimension is concerned with the method an adversary uses to distort the gradient value (cf. Algorithm REF ). We use a variety of state-of-the-art methods for distorting the computed gradients.
ALIE [1]} involves communication among the Byzantines in which they jointly estimate the mean \(\mu _i\)  and standard deviation \(\sigma _i\)  of the batch's gradient for each dimension \(i\)  and subsequently use them to construct a distorted gradient that attempts to distort the median of the results. Another powerful attack is Fall of Empires (FoE) [2]} which performs “inner product manipulation” to make the inner product between the true gradient and the robust estimator to be negative even when their distance is upper bounded by a small value. Reversed gradient distortion returns \(-c\mathbf {g}\)  for \(c>0\) , to the PS instead of the true gradient \(\mathbf {g}\) . The ALIE algorithms is, to the best of our knowledge, the most sophisticated attack in literature.
